A space-time adaptive low-rank method for high-dimensional parabolic partial differential equations

TL;DR

This work develops a space-time adaptive method for high-dimensional parabolic PDEs by coupling sparse time-wavelet expansions with per-time-index low-rank spatial representations. It provides a rigorous framework for computable a posteriori error bounds, convergence guarantees, and near-optimal computational complexity that avoids the curse of dimensionality under natural approximability assumptions. A key innovation is treating time separately via sparse wavelets while maintaining independent spatial low-rank representations for each active time index, together with diagonal space-time preconditioners to control ranks. Numerical experiments up to $d=128$ validate the approach, showing controlled residuals, modest rank growth, and practical scalability for high-dimensional diffusion problems.

Abstract

An adaptive method for parabolic partial differential equations that combines sparse wavelet expansions in time with adaptive low-rank approximations in the spatial variables is constructed and analyzed. The method is shown to converge and satisfy similar complexity bounds as existing adaptive low-rank methods for elliptic problems, establishing its suitability for parabolic problems on high-dimensional spatial domains. The construction also yields computable rigorous a posteriori error bounds for such problems. The results are illustrated by numerical experiments.

Figures (11)

• Figure 1: Example of a binary dimension tree $\mathbb{T}_d$ and its corresponding effective edges $\mathbb{E}_d$ in dimension $d=4$.
• Figure 2: Norms of computed error estimates and error bounds in dependence on the iteration number for the heat equation with source term, for $d=8,32,128$.
• Figure 3: Sum of the support and maximum rank per iteration versus the current residual norm estimate for the heat equation with source term, for $d=4,8,32,128$.
• Figure 4: Maximum ranks for each time index in dependence on the iteration number for the heat equation with source term, for $d=8, 32, 128$.
• Figure 5: Sum of one-dimensional support for each time index in dependence on the iteration number for the heat equation with source term, for $d=8,32,128$.

Theorems & Definitions (93)

• Theorem 2.1
• Remark 2.2
• Proposition 2.3
• Remark 2.4
• Definition 3.1
• Definition 3.2
• Proposition 3.3
• proof
• Lemma 3.4
• Definition 3.5